The Kelvin-Helmholtz (K-H) instability of a tangential discontinuity in a compressible plasma is reexamined in the linear magnetohydrodynamic (MHD) approximation. For fixed plasma conditions, two different kinds of surface waves (labeled F for fast and S for slow) may exist simultaneously with different tangential wave vectors kt. The surface waves can be excited only for a limited range of U, the relative flow speed of the plasmas on the two sides of the interface. Thus the instability requires Ucm Ui, but the slow wave growth rate is small in comparison with both &egr;F and &egr;i. Consequently, plasma compressibility is relatively ineffective in reducing the critical velocity for surface wave growth below that for an incompressible plasma. In particular, for the nominal conditions of the dayside equatorial magnetopause, S mode waves do not occur near the minimum UcS (i.e., with kt perpendicular to the magnetospheric magnetic field) because &egr;S →0 and UuS =UcS. On the terrestrial magnetopause the F(S) surface waves can couple a quasi-fast (quasi-slow) MHD mode in the magnetospheric plasma to either a quasi-fast or quasi-slow MHD mode in the magnetosheath. These waves decay as they propagate into the bounding plasmas. Consideration of the limit U→UuF reveals the significance of the upper critical velocity. In this limit, the phase velocity of the fast surface wave approaches the MHD fast mode speed in both bounding plasmas. The imaginary part of normal component of k vanishes and the unstable surface waves change to stable MHD waves that propagate away from the boundary without damping. The above points are discussed in relation to previous work on the K-H instability. Possible applications to observations in the terrestrial magnetosphere are mentioned.